SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
Popular matchings in the capacitated house allocation problem
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Bounded Unpopularity Matchings
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Dynamic matching markets and voting paths
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Journal of Combinatorial Optimization
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An instance of the popular matching problem (POP-M) consists of a set of applicants and a set of posts. Each applicant has a preference list that strictly ranks a subset of the posts. A matching M of applicants to posts is popular if there is no other matching M驴 such that more applicants prefer M驴 to M than prefer M to M驴. Abraham et al. (SIAM J. Comput. 37:1030---1045, 2007) described a linear time algorithm to determine whether a popular matching exists for a given instance of POP-M, and if so to find a largest such matching. A number of variants and extensions of POP-M have recently been studied. This paper provides a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a structure called the switching graph, a directed graph computable in linear time from the preference lists. We show that the switching graph can be exploited to yield efficient algorithms for a range of associated problems, including the counting and enumeration of the set of popular matchings, generation of a popular matching uniformly at random, finding all applicant-post pairs that can occur in a popular matching, and computing popular matchings that satisfy various additional optimality criteria. Our algorithms for computing such optimal popular matchings improve those described in a recent paper by Kavitha and Nasre (Proceedings of MATCH-UP: Matching Under Preferences--Algorithms and Complexity, 2008).